The sequences CN CV NC and CV are called
"bidents". They are a form of bonding
whereby we take a two-argument function
and fix the value of one of its arguments to get a
one-argument function.
However, there is a difference between bonding a dyadic
verb
(as in + & 2 for example) and bonding a conjunction.
With the conjunction, there is no need for a bonding
operator
such as &. We just write (" 0) with no intervening
operator.
The reason is that in the case of + & 2,
omitting the & would give + 2 which means: apply the
monadic case of + to 2, giving 2. However,
conjunctions don't have monadic cases, so the
bident (" 0) is recognised as a bonding.

Recall the "Under" conjunction &. from Chapter 08
whereby
f&.g is a verb which applies g to
its argument, then f then the inverse of g.
If we take f and g to be:

f =: 'f' & ,
g =: >

then we see that f is applied inside each box:

z

(f &. g) z

+-+-+-+
|a|b|c|
+-+-+-+

+--+--+--+
|fa|fb|fc|
+--+--+--+

Now, using the form CV, we can define an adverb EACH
to mean
"inside each box":

EACH =: &. >

f EACH

z

f EACH z

&.>

f&.>

+-+-+-+
|a|b|c|
+-+-+-+

+--+--+--+
|fa|fb|fc|
+--+--+--+

15.3 Compositions of Adverbs

If A and B are
adverbs, then the bident (A B) denotes an adverb
which applies A and then B. The scheme is:

x (A B) means (x A) B

15.3.1 Example: Cumulative Sums and Products

There is a built-in adverb \ (backslash, called Prefix). In the expression
f \ y the verb f is applied to successively
longer leading segments of y. For example:

< \ 'abc'
+-+--+---+
|a|ab|abc|
+-+--+---+

The expression +/ \ y produces cumulative
sums of y:

+/ \ 1 2 3
1 3 6

An adverb to produce cumulative sums, products,
and so on can be written as a bident of two adverbs:

cum =: / \ NB. adverb adverb

z =: 2 3 4

+ cum z

* cum z

2 3 4

2 5 9

2 6 24

15.3.2 Generating Trains

Now we look at defining adverbs to generate trains of
verbs, that is, hooks or forks.

First recall from Chapter 14
the Tie conjunction (`), which makes gerunds,
and the Evoke Gerund adverb (`: 6) which makes trains
from gerunds.

Now suppose that A and B are the adverbs:

A =: * ` NB. verb conjunction
B =: `: 6 NB. conjunction noun

Then the compound adverb

H =: A B

is a hook-maker.
Thus generates the hook * , that is
"x times x-1"

h =:

h 5

+-+--+
|*|
+-+--+

*

*

20

15.3.3 Rewriting

It is possible to rewrite the definition of a verb
to an equivalent form, by rearranging its terms.
Suppose we start with a definition of the factorial
function f. Factorial 5 is 120.

f =: (* ($: @:

The idea now is to rewrite f to the form
$: adverb, by a sequence of steps.
Each step introduces a new adverb.
The first new adverb is A1, which has the form
conj verb.